DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08
OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion of h Fourir ranform Pol-zro plo Propri of Laplac ranform Linariy Tim hifing Shifing in h -domain Tim caling Conjugaion Convoluion Diffrniaion in h im domain Diffrniaion in -domain Ingraion in h im domain Iniial and final valu horm Signal and Sym - Wk 0
THE LAPLACE TRANSFORM A h bginning of Fourir Tranform, h rpon of a linar im-invarian ym wih impul rpon h() o a compl ponnial inpu wa analyzd. y whr H H For =jw hi ingral corrpond o Fourir Tranform of h() For gnral valu of compl variabl, i i rfrrd a h Laplac ranform of h() h d Signal and Sym - Wk 0 3
THE LAPLACE TRANSFORM Th Laplac ranform of a gnral ignal () i dfind a No ha i a compl variabl and can b wrin a =+jw, wih and w ar ral and imaginary par For convninc, h Laplac ranform will b dnod a L{()} and h ranform rlaionhip bwn () and () will b dnod a L d Signal and Sym - Wk 0 4
THE LAPLACE TRANSFORM Whn =jw, h Laplac ranform bcom Fourir ranform jw Th Laplac ranform hav alo a rlaionhip vn i no purly imaginary Th Laplac ranform can b inrprd a h Fourir ranform of () - Dpnding on h valu of hi ignal may b dcaying or growing in im. jw d jw jw d jw Signal and Sym - Wk 0 5 d
EAMPLE a u a a u d a a d a R a 0 0 No ha for R{}<-a h ingral will no convrg and h rul will b infiniy. Thrfor, hr i a rquirmn for in ordr o guaran h convrgnc. Signal and Sym - Wk 0 6
EAMPLE a u a a u d a 0 a d a R a 0 For convrgnc in hi ampl, R{+a}<0 or R{}<-a Comparing prviou ampl, h algbraic prion for h Laplac ranform ar idnical. Howvr, hy diffr bad on hir convrg rgion. Signal and Sym - Wk 0 7
THE LAPLACE TRANSFORM In gnral, h rang of valu for which h Laplac ingral convrg i rfrrd a h rgion of convrgnc (ROC). A convnin way o diplay h ROC i o how in h compl -plan. Th horizonal ai i R{} and vrical ai i Im{} For ampl h hadd ara ar h ROC of prviou ampl. Im Im -a R -a R Signal and Sym - Wk 0 8
EAMPLE 3 Th ROC i h rgion whr boh rm convrg. For hi ampl, h fir rm will convrg for R{}>- and cond rm will convrg for R{}>- Combining hi wo ROC, h ROC i R{}>- 3 3 3 3 3 d u d u d u u u u Signal and Sym - Wk 0 9
THE LAPLACE TRANSFORM A n in prviou ampl, h Laplac ranform i a raio of polynomial in h compl variabl, o ha whr N() and D() ar h numraor polynomial and dnominaor polynomial. Ecp a cal facor, h numraor and dnominaor polynomial can b pcifid by hir roo and marking h locaion of h roo provid a picorial way of dcribing h Laplac ranform. Each roo of dnominaor polynomial i indicad wih and ach roo of numraor polynomial i indicad wih o N D Signal and Sym - Wk 0 0
THE LAPLACE TRANSFORM For ampl, h roo of dnominaor and numraor polynomial of ampl 3 ar hown. Im - - ` -plan R Th roo of numraor polynomial ar commonly rfrrd o h zro of (), inc for h valu of, ()=0 Th roo of dnominaor polynomial ar commonly rfrrd o h pol of (), inc for h valu of, () i infini Th rprnaion of () uing i pol and zro in h - plan i rfrrd o a h pol-zro plo o () Signal and Sym - Wk 0
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry : Th ROC of () coni of rip paralll o h jw-ai in -plan Th ROC of () coni of h valu of =+jw for which h Fourir ranform of () - convrg Thi condiion dpnd only on, h ral par of Propry : For raional Laplac ranform, h ROC do no conain any pol. Sinc () i infini a a pol, h Laplac ranform will no convrg a a pol. Signal and Sym - Wk 0
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 3: If () i a fini duraion ignal and aboluly ingrabl, hn ROC i h nir -plan. A fini duraion ignal i zro ouid an inrval of fini duraion. If () i aboluly ingrabl in hi fini inrval, muliplying i by dcaying or growing ponnial will alo b boundd. T T d T T d Signal and Sym - Wk 0 3
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 4: If () i righ idd, and if h lin R{}= 0 i in h ROC, hn all valu of for which R{}> 0 will alo b in h ROC. A righ-idd ignal i a ignal for which ()=0 prior o om fini im T. If a poin i in h ROC, hn all h poin o h righ of (all poin wih largr ral par) ar in h ROC. For hi raon, h ROC in hi ca i commonly rfrrd o a a righ-half plan. d T d Signal and Sym - Wk 0 4
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 5: If () i lf idd, and if h lin R{}= 0 in in h ROC, hn all valu of for which R{}< 0 will alo b in h ROC. A lf-idd ignal i a ignal for which ()=0 afr om fini im T. Th ROC i commonly rfrrd o a lf-half plan for hi ca. Signal and Sym - Wk 0 5
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 6: If () i wo idd, and if h lin R{}= 0 i in h ROC, hn h ROC will coni of a rip in h -plan ha includ h lin R{}= 0 A wo-idd ignal i a ignal ha i of infini n for boh >0 and <0. Thi yp of ignal can b dividd ino righ-idd R () and lf-idd L ()ignal. Th Laplac ranform will convrg for valu of for which h ranform of boh R () and L () convrg. Signal and Sym - Wk 0 6
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Im Im r R L R Im r L R Signal and Sym - Wk 0 7
REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 7: If h Laplac ranform () of () i raional, hn i ROC i boundd by pol or nd o infiniy. In addiion, no pol of () ar conaind in h ROC. Propry 8: If h Laplac ranform () of () i raional, hn if () i righ idd, h ROC i h rgion in h -plan o h righ of h righmo pol. If () i lf idd, h ROC i h rgion in h -plan o h lf of h lfmo pol. Signal and Sym - Wk 0 8
EAMPLE Conidr h pol-zro parn of h following Laplac ranform Im Im R R Righ-idd ignal Im Im Lf-idd ignal R R Two-idd ignal Signal and Sym - Wk 0 9
THE INVERSE LAPLACE TRANSFORM Th Laplac ranform of a ignal () wih prd a =+jw and in h ROC Signal and Sym - Wk 0 0 w w w w w w w w d j d j d j j j j by muliplying boh id h invr Fourir Tranform Uing
THE INVERSE LAPLACE TRANSFORM () can b rcovrd from i Laplac ranform valuad for a of valu of in h ROC wih fid and w varying from - o. j Thi quaion a ha () can b rprnd a a wighd ingral of compl ponnial. j d Signal and Sym - Wk 0
EAMPLE L uing u u L L u R R R h parial- fracion panion L R Signal and Sym - Wk 0
GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Th Fourir ranform of a ignal i h Laplac ranform valuad on h jwai. To dvlop a procdur o valua h Fourir ranform from Laplac ranform conidr a Laplac ranform wih a ingl zro [()=-a] L ay w wan o valua =. Th algbraic prion -a i h um of wo compl numbr and -a,which can b rprnd a vcor in h compl plan. Th vcor um of and a i a vcor from a o. Th lngh of hi vcor i h magniud of -a and h angl rlaiv o ral ai i h angl of hi compl numbr. If () ha a ingl pol [()=/(-a)], hn h rciprocal of h lngh of h vcor from pol i h magniud and h ngaiv of h angl i h ngaiv of h angl of h vcor wih h ral ai. Signal and Sym - Wk 0 3
EAMPLE Th, R Fourir Tranform i jw jw jw w jw an w jw Th lngh of hi vcor i h magniud Th ngaiv of h angl of h vcor i h angl. Signal and Sym - Wk 0 4
GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Gnral raional Laplac ranform To valua () a = Th magniud of ( ) i h magniud of h cal facor M im h produc of lngh of h zro vcor dividd by h produc of h lngh of h pol vcor. Th angl of ( ) i h um of h angl of h zro vcor minu h um of h angl of h pol vcor. If h cal facor M i ngaiv, an addiional angl of would b includd. If () ha mulipl pol qual o ach ohr or mulipl zro qual o ach ohr (or boh), h lngh and angl of h vcor from ach of h pol or zro mu b includd. Signal and Sym - Wk 0 5 P j j R i i M P j j R i i P j j R i i M
EAMPLE Conidr h following all-pa ym Th pha of H H H h frquncy rpon jw inc jw inc an w a jw an w a Signal and Sym - Wk 0 6
PROPERTIES OF THE LAPLACE TRANSFORM Propry Signal Laplac Tranform ROC R R R Linariy a b a b Tim Shifing 0 0 0 Shifing in h - Domain Tim Scaling a Diffrniaion in h Tim Domain A la R R R a Conjugaion Convoluion d Diffrniaion in h -Domain Ingraion in h Tim Domain if 0for 0 and if d 0 a Shifd vrion of R (i.., i in h ROCif 0 i in R) Scald ROC (i.., i in h ROC if / a i in R) R A la R R A la R d d d A la R R Iniial- and Final-Valu Thorm conain noimpul or highr - ordr ingularii a 0, hn 0for 0 and 0 lim lim ha a fini limi a, hn lim 0 R 0 Signal and Sym - Wk 0 7
THE INITIAL AND FINAL VALUE THEOREMS For h ignal ()=0, <0 and () conain no impul a h origin, i i poibl o calcula h iniial valu a approach zro from poiiv valu of. Thi horm i calld iniial-valu horm 0 lim If ()=0 for <0 and () ha a fini limi a, hn h final valu horm lim lim 0 Signal and Sym - Wk 0 8
EAMPLE Calcula h iniial-valu and final-valu for h following ignal. 5 3 4 4 0 3 5 lim lim 3 4 4 0 lim lim lim 0 0 3 5 0 3 4 4 0 Signal and Sym - Wk 0 9
SOME LAPLACE TRANSFORM PAIRS u Signal Tranform ROC All u R 0 u R 0 n R 0 u n n! n R 0 u n n! u R u R n u n! n R n R u n! n T T All cow 0u R 0 w0 inw 0u w0 R 0 w0 cow0 u R w0 inw 0 u w0 R w n n d un n d u u n im n n 0 All R 0 Signal and Sym - Wk 0 30